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In this paper we consider memoryless one-bit compressed sensing with randomly subsampled Gaussian circulant matrices. We show that in a small sparsity regime and for small enough accuracy $\delta$, $m\sim \delta^{-4} s\log(N/s\delta)$…

Compressed Sensing (CS) is an effective approach to reduce the required number of samples for reconstructing a sparse signal in an a priori basis, but may suffer severely from the issue of basis mismatch. In this paper we study the problem…

We describe a probabilistic, {\it sublinear} runtime, measurement-optimal system for model-based sparse recovery problems through dimensionality reducing, {\em dense} random matrices. Specifically, we obtain a linear sketch $u\in \R^M$ of a…

This article extends the concept of compressed sensing to signals that are not sparse in an orthonormal basis but rather in a redundant dictionary. It is shown that a matrix, which is a composition of a random matrix of certain type and a…

Atomic norm minimization is a convex optimization framework to recover point sources from a subset of their low-pass observations, or equivalently the underlying frequencies of a spectrally-sparse signal. When the amplitudes of the sources…

The techniques of low-rank matrix recovery were adapted for Quantum State Tomography (QST) previously by D. Gross et al. [Phys. Rev. Lett. 105, 150401 (2010)], where they consider the tomography of $n$ spin-$1/2$ systems. For the density…

The importance of regularization has been well established in image reconstruction -- which is the computational inversion of imaging forward model -- with applications including deconvolution for microscopy, tomographic reconstruction,…

Random sampling in compressive sensing (CS) enables the compression of large amounts of input signals in an efficient manner, which is useful for many applications. CS reconstructs the compressed signals exactly with overwhelming…

The problem of recovering signals of high complexity from low quality sensing devices is analyzed via a combination of tools from signal processing and harmonic analysis. By using the rich structure offered by the recent development in…

This paper studies the problem of reconstructing sparse or compressible signals from compressed sensing measurements that have undergone nonuniform quantization. Previous approaches to this Quantized Compressed Sensing (QCS) problem based…

An approximate sparse recovery system in $\ell_1$ norm consists of parameters $k$, $\epsilon$, $N$, an $m$-by-$N$ measurement $\Phi$, and a recovery algorithm, $\mathcal{R}$. Given a vector, $\mathbf{x}$, the system approximates $x$ by…

One-bit compressed sensing (1bCS) is an extreme-quantized signal acquisition method that has been intermittently studied in the past decade. In 1bCS, linear samples of a high dimensional signal are quantized to only one bit per sample (sign…

Iterative hard thresholding (IHT) and compressive sampling matching pursuit (CoSaMP) are two types of mainstream compressed sensing algorithms using hard thresholding operators for signal recovery and approximation. The guaranteed…

As shown in [Blumensath and Davies 2009, Baraniuk et al. 2010], signals whose wavelet coefficients exhibit a rooted tree structure can be recovered using specially-adapted compressed sensing algorithms from just n=O(k) measurements, where k…

Compressed sensing is a signal processing scheme that reconstructs high-dimensional sparse signals from a limited number of observations. In recent years, various problems involving signals with a finite number of discrete values have been…

In this manuscript, we analyze the sparse signal recovery (compressive sensing) problem from the perspective of convex optimization by stochastic proximal gradient descent. This view allows us to significantly simplify the recovery analysis…

This paper investigates total variation minimization in one spatial dimension for the recovery of gradient-sparse signals from undersampled Gaussian measurements. Recently established bounds for the required sampling rate state that uniform…

Let A be an M by N matrix (M < N) which is an instance of a real random Gaussian ensemble. In compressed sensing we are interested in finding the sparsest solution to the system of equations A x = y for a given y. In general, whenever the…

We propose a robust and efficient approach to the problem of compressive phase retrieval in which the goal is to reconstruct a sparse vector from the magnitude of a number of its linear measurements. The proposed framework relies on…

We obtain an improved Sobolev inequality in H^s spaces involving Morrey norms. This refinement yields a direct proof of the existence of optimizers and the compactness up to symmetry of optimizing sequences for the usual Sobolev embedding.…